Wire tension at spooling is determined by the passive machine elements after the last machine component that eliminates tension (Say a drawing machine dry capstan.). Such items as pulley bearing friction, guide surface friction and possibly windage in the the manufacture of magnet wire are all passive machine elements. The spooler itself, if in torque control, could also increase wire tension by pulling too hard on the wire after the last machine element that eliminates tension.

A dancer for a speed controlled spooler could also add tension. The weight of the dancer divided by the number of wires supporting the dancer would be the tension added along with the pulley friction. Added to this would be +ma or -ma depending upon which way the dancer was accelerating at any particular point in time.

The basic things that affect winding tensions are the same that affects any wire winding process. A more important question might be how does winding tension affect magnet wire (winding wire or enamelled wire). Magnet wire or winding wires (called either because they are wound into coils that creat magnetic fields.) are annealed during the enamelling process and as a result the wire tensile strength is reduced and the wire has greater ductility. This makes the wire easier to stretch or break.
When enamelling wire, the line speed is set and should not vary because if it did it would affect the quality of the cure and therefore the electrical properties of the wire. Spools used for magnet wire may have a straight barrel or a tapered barrel. As the spool fills up its rpm’s decrease but the winding tension remains constant. With tapered spools, the rpm’s are higher at the small end of the spool than at the larger but if the take-up is working right, the winding tension is constant at any barrel diameter.
Problems that might occur during the enamelling and spooling process include:
• Poor spooling at the flanges (over or under fill) basket weave/loose spooling causing the wire to get trapped and then breaking when being de-reeled by the customer.
• Under cured wire or sticky wire being tightly wound and sticking and breaking when de-reeling.
• Using spools that are not designed for the take up.
• Excess tension stretching the wire. Stretching even a fraction of an awg will change the resistance of the wire and make it out of spec.
• Traverse speed too fast or too slow can cause soft spooling or excessive overlap trapping the wire

Regarding above questions:
1- the type of lubricant is often dictated by the customer. Applied lubricants are either a mineral oil or a prepared wax liquid.
Some enamel is marketed as being low friction and are applied as an overcoat however there are sometimes thermal and electrical property issues. If you are considering enamel with some form of silicon oil in it -DON'T!! This sounds like a good idea but the silicon will ruin your oven's catalysts.
2-The amount to apply -as little as necessary to meet customer requirements.
3-Spring back, elongation, LSE, are all dictated by the magnet wire specs. You set your process so that the wire will meet the specs.
4. Pitch -you just have to adjust it until you have a pitch that provides the best spooling. Obviously it will vary with wire size, spool size and type. With new equipment, the traverse rate is tied to the rpm of the spool. As the spool fills its rpm decreases and the traverse rate should also change.